Question

define an $upno$ to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. similarly, define a $downno$ to be a positive integer of $2$ or more digits where the digits are strictly decreasing moving left to right. for instance, the number $258$ is an upno and $8620$ is a downno. let $u$ equal the total number of $upnos$ and let $d$ equal the total number of $downnos$. what is $|u-d|$?

Answer 1

An 'upno' is a positive integer with strictly increasing digits moving left to right, while a 'downno' is a positive integer with strictly decreasing digits moving left to right. We need to find the difference between the total number of upnos and downnos. The result is 0.

Step-by-step explanation:

An 'upno' is a positive integer with strictly increasing digits moving left to right. For example, 258 is an upno. On the other hand, a 'downno' is a positive integer with strictly decreasing digits moving left to right. For example, 8620 is a downno. We need to find the difference between the total number of upnos and downnos.

Let's count the number of upnos first. To have a 2-digit upno, we can choose any two unique digits from 1 to 9 (9 choices) and arrange them in increasing order, which gives us 9C2 = 36 ways.

Similarly, for a 3-digit upno, we can choose any three unique digits from 1 to 9 (9 choices) and arrange them in increasing order, which gives us 9C3 = 84 ways. Continuing this pattern, the total number of upnos is 36 + 84 + 126 +...+ 252 = 756.

Now, let's count the number of downnos. To have a 2-digit downno, we can choose any two unique digits from 0 to 8 (9 choices) and arrange them in decreasing order, which gives us 9C2 = 36 ways.

Similarly, for a 3-digit downno, we can choose any three unique digits from 0 to 8 (9 choices) and arrange them in decreasing order, which gives us 9C3 = 84 ways.

Continuing this pattern, the total number of downnos is 36 + 84 + 126 +...+ 252 = 756.

Therefore, |u-d| = |756-756| = 0.